Hundreds of millions of years ago there were dinosaurs, and they kept laying dinosaur eggs. But slowly they evolved into birds. So one day an animal that was not-quite-a-chicken layed a not-quite-a-chicken egg, but it hatched into a chicken!

it looks like Adam and myself use the same logic to come to different conclusions…

I think the answer is “the egg”. Dinosaurs were born from eggs but there were no chickens around back then. So eggs were first in general.

If you mean the more specific case of whether the “chicken-egg” was there first, I have to say it’s still the egg, because:
Evolution happens through tiny mutations in the offspring, always a minuscule one at a time. So even for humans there would have been a step where one time the mother would not be considered human but then the child was mutated in such a way that it was what would become known as “human”.
The same applies to the chicken. One day there was something that was very similar to a chicken but not a chicken (a dino-chick?) that laid an egg. In that egg was a slightly mutated baby that, once born, would be what became known as “chicken”. So the egg was first!

Isn’t that the question on all our lips, all of time???
Well maybe the egg came from from a species that was around before chickens and then these species eventually evolved to become what we know today as chickens.

Then it’s a continuum of eggs and mutated animals stretching all the way from dinosaur to chicken. But eggs and quasi-chickens are quantized so there must be one that is the turning point from dinosaur to chicken. But does it fall on the chicken side or the egg side?